On the Variability of Drop Size Distributions over Areas

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On the Variability of Drop Size Distributions over Areas
A. R. JAMESON
RJH Scientific, Inc., El Cajon, California
M. L. LARSEN
College of Charleston, Charleston, South Carolina
A. B. KOSTINSKI
Michigan Technological University, Houghton, Michigan
(Manuscript received 6 September 2014, in final form 25 December 2014)
ABSTRACT
Past studies of the variability of drop size distributions (DSDs) have used moments of the distribution such
as the mass-weighted mean drop size as proxies for the entire size distribution. In this study, however, the
authors separate the total number of drops Nt from the DSD leaving the probability size distributions (PSDs);
that is, DSD 5 Nt 3 PSD. The variability of the PSDs are then considered using the frequencies of size [P(D)]
values at each different drop diameter P(PD j D) over an ensemble of observations collected using a network
of 21 optical disdrometers. The relative dispersions RD of P(PD j D) over all the drop diameters are used as
a measure of PSD variability. An intrinsic PSD is defined as an average over one or more instruments excluding zero drop counts. It is found that variability associated with an intrinsic PSD fails to characterize its
true variability over an area. It is also shown that this variability is not due to sampling limitations but rather
originates for physical reasons. Furthermore, this variability increases with the expansion of the network size
and with increasing drop diameter.
A physical explanation is that the network acts to integrate the Fourier transform of the spatial correlation
function from smaller toward larger wavelengths as the network size increases so that the contributions to the
variance by all spatial wavelengths being sampled also increases. Consequently, RD and, hence, PSD variability will increase as the size of the area increases.
1. Introduction
Ever since the first reports of the frequency distribution of drops of various sizes [i.e., drop size distributions
(DSDs)] (Laws and Parsons 1943; Marshall and Palmer
1948; Best 1950), there has been a growing appreciation
of their variability particularly with regard to different
meteorological conditions. For decades, a great deal of
effort has been spent characterizing DSDs of all kinds,
so much so that there are now hundreds and possibly
thousands of expressions. Moreover, to a large extent,
the appearance of the field of radar meteorology arose in
response to DSDs because the integral properties of the
Corresponding author address: A. R. Jameson, 5625 N. 32nd St.,
Arlington, VA 22207-1560.
E-mail: [email protected]
DOI: 10.1175/JAS-D-14-0258.1
Ó 2015 American Meteorological Society
radar reflectivity factor Z and the rainfall rate R could be
directly connected, thus making radar a potentially
practical instrument for rapidly measuring rain over
large areas. While this potential is still being explored
today, the variability of the DSDs not only has led to
a vast family of Z–R relations, but the DSDs have also
produced families of parameterizations that relate the
variables of the DSDs to their integral properties. Unfortunately, such parameterizations, by their very nature, tend to minimize a full appreciation of the true
variability of the underlying DSDs.
This is not to say that such variability is not appreciated particularly as it impacts radar reflectivity factor,
beginning with Wexler (1948) and Jones (1956) up to the
present (e.g., Islam et al. 2012a,c), and rainfall spatial
distributions (e.g., Lee and Zawadzki 2005; Lee et al.2009;
Tokay and Bashor 2010; Jaffrain and Berne 2012a). These
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past studies, however, only use variability of the moments
of a size distribution as a proxy for the variability of the
entirety of the DSDs themselves.
Furthermore, more recently Jaffrain and Berne (2012b)
have proposed a decreasing stretched exponential function to describe the spatial correlation for DSD integrated
parameters for detector separations greater than 100 m.
What this means with regard to the high temporal and
spatial resolution of the DSDs themselves is not clear,
however. As important as such studies are for many
practical purposes, so far no one has considered looking
directly at the variability of the DSDs at each drop size on
scales less than 100 m as we do in this study.
As just mentioned, DSD variability has often been
evaluated using integrated quantities such as the mean
diameter. This is inadequate, however. Integrated variables depend not only on the form of the DSD but also
on the limits of the integration. Thus, the same value of
the integrated variable can be produced by a wide variety of different DSDs. Consequently, the variability of
integrated quantities fails to capture the total variability
of the DSDs. So what do we mean here by DSD variability? Normally, one has access to time series measurements using a single instrument, and normally
elements of the time series are averaged to reduce
fluctuations arising as a result of a variable number of
drops at all different sizes. Hence, there is variability due
to sampling and that due to real physics. We will show
below that it is the latter that dominates over a network.
Specifically, a drop size distribution can be described
as the product of the total number of drops Nt times
the frequency distribution of drops with size P(D) as
argued in Kostinski and Jameson (1999); that is,
N(D)dD 5 Nt P(D)dD, where P(D)dD is the probability
of finding a drop size between D and D 1 dD. That is,
DSD 5 Nt 3 PSD, where PSD 5 P(D). While Nt plays
a critical role in the variability of the rainfall rate and
other quantities (e.g., Jameson and Kostinski 2001;
Jameson 2015; Jameson et al. 2015), the focus here is on
the variability in the frequencies of different sizes of
drops—that is, on the variability of P(D). Specifically,
one takes the observed drop counts at the different size
bins using a disdrometer, whether impact or optical, and
normalizes the count at each drop size by Nt to yield the
estimate of P(D), the frequency or probability distribution of drop sizes. From observation to observation,
P(D) 5 PD varies at each D. We then have a probability
distribution of the different values of PSD at each D
denoted by P(PD j D). Thus, the variability of the PSD is
equivalent to the variability of P(D) at each size over all
sizes over an ensemble of measurements. Hence, in this
work, when we speak of the variability of the drop size
distribution, we mean the spread or relative dispersion
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in the values of P(PD j D) at all the different drop sizes
relative to their mean P(D). Later, we also provide
a measure over an entire network of instruments.
Below we describe a network of 21 optical disdrometers used to explore the variability of PSD. There
are two ways of looking at these data. The first is to
combine all the observations so that each instrument
contributes to a distribution of P(D) at each size. What,
then, is the average PSD regardless of the location of the
instrument? In effect, this excludes all P(D) 5 0 that
contribute no information. [For example, in the extreme
case when all P(D) 5 0, there is nothing to talk about.]
Then at each drop size, there will be an ‘‘intrinsic’’ relative dispersion of P(PD j D) reflecting the variability of
P(D) at each D excluding all zeros.
The second way of considering the data is to ask,
‘‘What is the average drop size distribution over the
area of the network each minute?’’ That is, what happens when one takes an intrinsic PSD and distributes it
spatially over an area? Now, the occurrences of P(D) 5
0 are meaningful since voids can occur within the area
encompassed by a network. Such variability impacts
the answer to that question. This distinction, while
seemingly academic, can strongly influence the
network-relative dispersions of P(PD j D) at each size
and, consequently, the potential variability of the
network-average PSD. This will be discussed further
below.
While seemingly pedantic, this distinction is important when one wants to consider rainfall over an area
because the intrinsic P(D) alone does not fully account
for the effects of spatial variability particularly the spatial voids, which may appear at different sizes at different locations and times. Obviously, integration time will
influence but not eliminate the occurrences and locations of these voids since they are not all due to sampling
fluctuations. To provide an illustration of such spatial
variability, Fig. 1 shows the PSDs [P(D)] observed every
minute by two detectors over a 440-min rain event
consisting of both intense rain from thunderstorms and
light rain from subsequent anvil containing a few weak,
transient intense elements discussed further below.
Even though the two identical instruments are separated
by only 1.93 m, the profiles of P(D) can still be significantly different (aside from instrument differences) as
indicated in Fig. 1c and as discussed further later. These
differences (Fig. 1a) between the two PSDs were not
influenced by any statistically significant bias in Nt between these two detectors for these data (Fig. 1d).
In the next section, we describe the network in greater
detail, provide a few definitions, and establish the connection between the relative dispersions of P(PD j D) at
each D and the variability in both the intrinsic and the
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FIG. 1. A contour plot of log10[P(D)] over the entire 440-min rain event for two optical disdrometers in the network (see Fig. 2):
(a) optical disdrometer A and (b) optical disdrometer B, only 1.93 m away from A. (c) A plot of the ratio of the two P(D) highlighting
differences between the two detectors. (d) The scatterplot between Nt for detectors A and B showing no bias.
network-average PSDs. Data analyses and results will
then follow.
2. Preliminary considerations
a. The network of detectors
The network consists of 21 Thies Laser Precipitation
Monitors (LPMs), in conjunction with a Joanneum
compact two-dimensional video disdrometer (2DVD).
The array is located at historic Dixie Plantation near
Hollywood, South Carolina; this property (owned by the
College of Charleston Foundation) is used for a variety
of ecological, educational, and research purposes. The
site is located at 328440 2600 N, 808100 3600 W.
The instrument layout is shown in Fig. 2 and was designed to develop a dense network with distinct spatial
separations. This layout contrasts with the usual grid
setup that collects a lot of information at only one
a particular separation distance but then abandons
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FIG. 2. A schematic of the layout of the LPM (optical disdrometers) network, with the letters referencing specific instruments as further discussed in the text. The 2DVD is indicated
by the square. The origin of the network is taken to be at detector A
in the upper-left corner. The arms refer to specific linear arrays of
instruments.
information on many other scales. By using logarithmic
spacing, however, spatial scales from approximately 1 to
100 m can be explored simultaneously. With the addition of the compact 2DVD (which is capable of resolving
spatial locations of drops to within less than 1 mm), this
array then allows us to investigate rainfall spatial variability through five orders of magnitude, most of which
have not been extensively explored in past studies.
Each LPM was calibrated in a separate indoor laboratory and moved to the field site. This study uses data
taken by 19 operational LPMs (LPMs L and M had
glitches in the wiring that were not rectified until early
December).
The Thies LPM instruments were characterized in
detail by de Moraes Frasson et al. (2011) and have been
used in a number of other studies including Brawn and
Upton (2008) and Fernández-Raga et al. (2009, 2010).
Optical disdrometers are well recognized as useful tools
for characterizing drop size distributions (e.g., LöfﬂerMang and Joss 2000; Tokay et al. 2001). It is not likely
that the results presented here depend upon the type of
disdrometer (Islam et al. 2012b). The instruments are
infrared occlusion instruments that can be run in several
separate modes; for this study, the instruments were run
in their default mode associated with 1-min integrations;
in this mode, the device reports a spectrum each minute
indicating how many droplets were detected in each of
22 disjoint drop size bins and 20 disjoint velocity bins
(thus, each drop is characterized as belonging to one of
440 different categories). The known issues associated
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with particle sizing were mitigated to the greatest degree
possible by verifying consistent performance in the
laboratory before deployment and using identical instruments throughout the array. Moreover, measurements by all of the instruments were compared to
minimize the inclusion of questionable behavior.
The devices are naturally synchronized; all devices
were turned on simultaneously since they are powered
with the same power supply. Although these devices are
intended to be very low maintenance, the acquisition of
data is reset every week to ensure minimal temporal
drift; empirical estimation from this process suggests
a relative shift of less than 1 s week21 among all detectors. During the weekly maintenance activity, the
devices are confirmed to be level, free of debris/insects,
and recording properly.
The instruments are placed along three arms: two
being orthogonal and the third bisecting the right angle.
Eight instruments are spaced logarithmically out to
100 m along two of the arms, but the third arm, having
seven instruments, only extends 52 m. This spacing provides increasing spatial resolution toward the network
origin. One-minute drop counts over 22 size bins are recorded every minute for all the instruments so that we can
estimate the PSDs for all the detectors every minute.
From this entire grid extending out to 100 m, grids of
other sizes can be constructed as well. In this study we
consider a 2-m grid (detectors A, B, H, Q), a 4-m grid
(A, B, H, Q, C, J, R), and a 7-m grid (A, B, H, Q, C, J, R,
D, K, S) as well as the full grid containing all of the instruments out to 100 m. Other grids can, of course, be
constructed, but because we wished to look at the
changes associated with the expansion of a network size
rather than its relocation, we have the larger grids containing the smaller subgrids.
b. The effect of a network on the relative dispersion
As mentioned above, in this work the variability of
a PSD is characterized by the relative dispersion of the
distribution of P(D) at each drop size. We denote this
distribution of P(D) at each size by P(PD j D) where the
vertical bar denotes ‘‘at D.’’ In Fig. 3 we illustrate one
distribution P(PD j D) of values observed at the indicated drop size (black line). In this case, there is near
full occupancy of the network since P(PD j D) at zero is
very small. That is, during the 440 one-minute observations over all 19 detectors, the average P(PD j D) at
this size was almost never close to zero. In part that is
because these small sizes are ubiquitous, but for larger,
scarcer drops, that is not necessarily true. To illustrate
a more porous rain—that is, rain with voids at that size—
we take this observed distribution and modify it to
simulate only 60% occupancy or, to put it another way,
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As «/0 and defining the fraction of detectors occupied
as f 5 (M 2 m)/M,
sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
1 2
12f
RD 5
R 1
, where 1/M # f # 1. (3)
f Di
f
FIG. 3. The probability distribution by P(PD j D) of P(D)
showing the observations (black) and the results had 40% of the
network been devoid of drops of this size (red). The mean m and
standard deviation s illustrate that the relative dispersion would
have increased RD 5 s/m in this latter case.
40% of the time there were very few if any drops of this
size in the entire network so that the network average
was smaller than the smallest bin size of P(D) used to
compute P(PD j D). As Fig. 3 illustrates (red lines), both
the mean m and the standard deviation s are altered with
the latter increasing and the former decreasing. Consequently, the relative dispersion, RD 5 s/m, increases
with decreasing occupancy. Thus, in general for each
size, RD over a network will be larger than RDi [i.e., the
value of the relative dispersion of P(PD j D) for the intrinsic PSD at size D].
The results depicted in Fig. 3, however, can be expressed explicitly. Consider a network having M detectors.
Suppose that m of these have a very small P(D) # «, and
suppose that the other (M 2 m) values have an average
P(D) 5 x and P(D)2 5 x2 . Then it follows that
m5
X2 5
m« 1 (M 2 m)x
M
and
m«2 1 (M 2 m)x2
,
M
(1)
where m is the mean and X 2 is the average squared
value. It then follows that
R2D 5
5
X2
21
m2
Mm«2 1 M(M 2 m)x2
m2 «2 1 2m«(M 2 m)x 1 (M 2 m)2 x2
2 1.
(2)
Consequently, as f /1, RD / RDi. On
the other hand,
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
as f / 1/M, its smallest value, RD / M 2 1 since for
one grid value RDi / 0 so that RD becomes only
a function of the total number of detectors. This relation
is plotted in Fig. 4 where it is clear that RD . RDi for all
f , 1.
Note that the M detectors can be distributed over an
area of any size. The effect of different sizes of areas,
then, is expressed through f, which depends on the
sparseness of the rain. As we shall see, it also depends
upon drop size so that, in reality, RD never approaches
RDi simultaneously over all the drop sizes. Because the
relative dispersion is a measure of the spread of probabilities around the average P(D) at each size, this means
that it is less likely that P(D) will simultaneously be near
their mean (intrinsic) values at all D. In that sense, the
intrinsic PSD is less likely to be realized spatially over
a network so that simply using a mean PSD and its associated RDi is misleading particularly if measured by
one instrument over time.
3. Data and analyses
a. An initial observation
With these tools, we begin the analyses of a rain
event consisting of both a more intense component
and a lighter component that began at 1645:00 UTC
23 Nov 2013 (Saturday) and lasted for 440 min with
maximum rainfall rates approaching 250 mm h21. To
get a basic feeling for this rain event, the rainfall rate
averaged over the entire network is illustrated in
Fig. 5. Only 19 of the 21 detectors were operational
during this event (i.e., excluding detectors L and M as
shown in Fig. 2) so that this average is only over those
disdrometers. While for the most part we consider all
440 min, we also analyze two selected time periods as
indicated with the red portion denoting the more intense rain and the green denoting the lighter rain. Each
is 120 min long.
Given the above discussion, then it should come as no
surprise that P(D) could vary considerably from minute
to minute over a network. Indeed, Fig. 6a illustrates the
differences even between neighboring detectors A and
B separated only by 1.93 m as previously shown in
Fig. 1c. While the green areas denote P(D) values that
are nearly the same for the two detectors, detector A
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FIG. 5. The network-averaged rainfall rate for the (a) intense rain
and (b) light rain as discussed in the text.
FIG. 4. Contours of RDi as a function of RD and the fraction of the
network occupied by the drops. Note the increase in RD for a fixed
RDi as the fraction filled decreases.
often showed greater P(D) at the larger sizes, with the
opposite being true at the smaller diameters. This, of
course, could be due to subtle differences between the
instruments, both of which were calibrated in the laboratory before deployment. However, even after
computing the average P(D) over the 2-m grid containing four detectors all lying within 1.93 m of detector
A, there is still considerable variability in time at larger
sizes as shown in Fig. 6b. One can also sense the increasing variability (relative dispersion) with increasing size by the spikiness. The same variability and
trends with increasing diameters in the average P(D)
persists even when considering the entire 100-m network as illustrated in Fig. 6c. Yet all three examples
show persistence in many of the overall structures in
time regardless of the size of the network suggesting
that the variability is real.
Indeed, these points are well illustrated in Fig. 7. With
the exception of the very largest drop sizes, it appears
that the mean values of P(D) (solid lines) are strikingly
similar regardless of the size of the network. What is
clearly different are the relative dispersions (dotted–
dashed lines) showing the variability of a 1-min PSD
over different sizes of networks. As one would expect,
while RD for a single instrument can be quite substantial
(magenta line), this potential variability is significantly
reduced after averaging over several detectors in a network. For example, the peak RD of 8.2 observed
pﬃﬃﬃﬃﬃ at the
largest sizes by one detector is reduced by 1/ 19 to 1.92
as observed over all 19 instruments. This would seem to
suggest that the variability is entirely due to sampling.
However, that turns out not to be correct because for
over a 2-m network of M 5 4 detectors, RD actually
decreases even more than for 19 detectors. Sampling
would suggest RD should have increased to 4.1, not decreased to 0.8. Something else is happening. Given the
discussion concerning the fractional occupancy effect,
this could be explained simply if the fractional occupancy f increased with decreasing network size as (3)
requires. Then for the same RDi, it would be possible for
RD to decrease even further even though there are fewer
instruments. However, that is not what is happening to
f 5 (M 2 m)/M, where m is the number of ‘‘no count’’
detectors out of M instruments in the network.
Instead, we see in Fig. 8 that the fractional occupancy
is actually increasing with increasing network size. This
would imply that RD should decrease toward RDi as the
network size increases. Instead, Fig. 7 shows it increasing with increasing network size. Hence, we must
conclude that RD is actually increasing with increasing
network size for a different reason. In other words, as
a network extends over a larger area, the natural varipﬃﬃﬃﬃﬃ
ability in P(D) actually increases in spite of any 1/ M
instrument effect.
b. A physical interpretation
This can be understood in terms of the spatial
spectrum of drop concentrations or counts n as discussed and calculated in Jameson et al. (2015). Because n 5 Nt 3 P(D j D), the spectrum of n is identical
to that for P(D j D). That is, at each drop size there is
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FIG. 6. Contour plots of P(D) for (a) detector A as in Fig. 1a, (b) 10log10P(D) over four detectors within a radius of 1.93 m from detector A,
and (c) 10log10 of the network-average P(D) over the entire network illustrating the effect of network (area) size.
a mean concentration, but the variance of the concentration produces the variance in P(D) as well as in
RD. Given a reasonable number of detectors to adequately sample over the area, the physical reason for
the increased variance is that as the area sampled by
a network increases, the contributions to the variance
by all spatial wavelengths in the part of the spectrum
being sampled also increases. If we just consider
a simple sinusoidal wave having amplitude ai modulating around the mean concentration, the variance
will be proportional to jaij2. As the dimension of the
area increases, the net variance will be proportional to
the sum of all the amplitudes weighted by their spectral contributions. Hence, as the dimension of the area
increases, the net variance will increase monotonically. Consequently, RD will increase as the size of
the area increases at least up until a wavelength of
about 1/ 2 the characteristic dimension of the area if
such wavelengths are present. While this cutoff is not
rigid, longer wavelengths contribute mostly to the
mean value, which is subtracted when computing the
variance. Another way to think of it is that the network acts to integrate the Fourier transform of the
spatial correlation function from the highest (smallest) toward lower (longest) frequencies (wavelengths)
as the dimension of the network increases. Hence, this
finding is quite general and not peculiar to this particular set of measurements.
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FIG. 7. The intrinsic P(D) averaged over different sizes (areas) of
networks (solid lines) and the observed total RD for the different
networks as functions of drop size over the 440 min of observations.
Note the increase in RD with increasing network size (area) and the
similarities among the average PSDs.
Moreover, this understanding can also explain why
RD increases with increasing drop size for a fixed area
covered by a network. The spatial distribution of drops
is patchier (i.e., the drops occur more locally with more
and larger gaps in between) as drop size increases (e.g.,
Jameson et al. 2015). Characterization of this patchier
spatial distribution, then, requires greater spatial
spectral widths as is well known from Fourier transform theory. Consequently, for the reasons given
above, the variances must also increase as drop size
increases. This effect is further amplified by the tendency toward lower mean P(D) with increasing drop
size so that the net effect is an increase in the relative
dispersions as the drops become larger just as is
observed.
In turn, this implies that the total variability of the
distribution over a network should also increase. One
measure of this total variability is the sum of the relative dispersions over all drop sizes as illustrated in
Fig. 9. Obviously, the total variability increases
monotonically with increasing dimension of the network. While this can be well fit parametrically by
a power law, the ranges of the variables are too small to
allow any physically reliable interpretation, and one
should certainly not extrapolate to dimensions greater
than those of the observations. This also suggests that if
one had access to the full relevant spatial spectrum (i.e.,
Fourier transform of the spatial correlation function), in
a particular situation, it might be possible to calculate
how the total variability would scale with the size of an
area. Unfortunately, such information is not readily
available.
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FIG. 8. Plots of the total fraction of the network filled during the
entire period of observation as a function of drop diameter. Note
that the fraction occupied actually increases as the network size
(area) increases so that it cannot explain the observed increase in
RD with increasing network size (area).
c. Further analyses
Naturally, as one would expect, this same effect appears to be true regardless of the meteorology as illustrated in Fig. 10, where we subdivide the data into more
intense or lighter rain. The same network size (area)
FIG. 9. The total relative dispersion (total variability) of the size
probability distribution over the network plotted as a function of
network dimension calculated by summing over all drop sizes.
While this can be well fit parametrically by a power law, the ranges
of the variables are too small to allow any physically reliable
interpretation.
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FIG. 11. Plots of the average intrinsic PSD and that observed
over the entire network. The red-filled area illustrates the spread
associated with the intrinsic PSD. Note that the mean curve for the
entire network falls outside this region for drops larger than 4-mm
diameter.
FIG. 10. As in Fig. 7, but for (a) the more intense rain and (b) the
more stratiform rain noted in Fig. 5.
effect on the relative dispersion is apparent in both instances, as would be expected given the above discussion. There are some differences between the lighter and
more intense rain, of course, as one would expect for
different spatial spectra.
Clearly and, in retrospect, not surprisingly, these results imply that there can be a significant scatter of observed PSDs about any overall mean PSD. This is well
illustrated in Fig. 11, where the 2s bounds about the
intrinsic average PSD are plotted. While this bounded
area appears to be a reasonable size, the behavior of the
network-average P(D) (dotted–dashed line) at larger
drop sizes suggests that such a conclusion fails to capture
all of the variability over an area (network).
Indeed, just knowing a mean intrinsic PSD has its
limitations with regard to the anticipated variability over
an area. That is, one should expect that fluctuations in
integrated variables such as R, Z, and the kinetic energy
available for soil erosion should exceed what would be
implied if one just considered fluctuations in the mean
intrinsic PSD. That is, the natural variability of these variables is larger when the intrinsic PSD is spread out over
areas because of the effects of spatial variability. Unfortunately, this is typically ignored in such calculations.
So what is more representative of the actual variability
over a network (area) since Fig. 11 is incomplete? To
reduce the effects of randomness in the sampling, one
way to address this question is to consider time-average
observed network PSD. Here we will consider 20-min
network-average PSD. For the 440-min period of observations, this provides only 22 observed PSDs over the
network, however. While useful, such a limited set is
unlikely to capture the full potential real variability implied by the set of observations. To expand the size of
a set of potential PSDs, we use the observed network time
series of drop counts to define the distribution of mean
drop counts using Bayesian analyses (Jameson 2007). In
addition, the network-average temporal correlation
functions at each size can be computed. These two components are then used to expand the size of the observed
set of data in a manner consistent with the observed
counts at all the drop sizes and with the observed correlation functions as explained in detail in Jameson (2015).
While the reader is referred to these latter references
for a more complete discussion, we briefly describe the
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procedure here. For each observed count of particles of
a specified size, the application of the Bayesian approach under an assumption of counting statistics
(Poisson or Gaussian, for example) produces a distribution of mean values of counts per unit observation interval. The most frequent mean values then emerge
naturally as the most likely (ML) values surrounded by
an estimate of the entire probability density function
(pdf) of these mean values. Zeroes are included in this
distribution to account for the frequency of data voids at
each size during the observations.
The correlation among drops of the same size is described by the appropriate correlation function, usually
an exponential having the correct 1/e correlation length,
where e is the Euler number. In this work the cross
correlation among drops of different sizes is not directly
considered but instead arises naturally by applying the
autocorrelation function for each different drop size to
the same field of random numbers as described and
demonstrated in Jameson (2015). This correlation of
counts can be interpreted as a description of the meteorological ‘‘structure,’’ which, in turn, is a reflection of
whatever physical processes produced it. A detailed
knowledge of the meteorology is not required to use the
correlation information. At each drop size, these correlation functions are then applied to an uncorrelated
string of unit variance numbers uniformly distributed
over [0, 1]. These correlated numbers are then used to
generate a string of correlated drop counts at each size
using the observed distributions of the mean counts and
using the copula statistical technique. This technique is
described in some detail with the appropriate references
in Jameson (2015) and will not be repeated here. It
should be noted that different random strings yield different sets of counts. This approach, then, makes it
possible to form additional PSDs not actually observed
but still consistent with the actual measurements. In
effect, this approach provides a much fuller expression
of all of the information contained within the observed
correlations and drop counts than is given by just the
realizations actually observed.
Using this approach for two different strings of random numbers, two sets each of 40 000 one-minute network counts at each drop size bin are generated. This
now gives us 4000 additional 20-min PSDs that are
consistent with the original set of data. In Fig. 12, these
PSDs (gray) are plotted along with the observed (blue)
22 twenty-minute-average PSDs. Obviously, both the
observed and simulated network 20-min-average PSDs
show considerably greater variability than the RDi associated with the intrinsic PSDs would imply. This is
important because it means that collecting measurements over an area does not necessarily lead to less
1395
FIG. 12. As in Fig. 11, but with overlays of observed 20-min PSDs
(blue lines which become black in the red areas) and simulated
20-min-average PSDs (gray) derived from the expanded dataset as
discussed in the text. Note the much greater network variability
compared with that for the intrinsic PSD.
spread in the estimated PSDs even though many more
instruments are included.
Moreover, this variability seems to increase significantly beyond about 1.5-mm diameter. To show this
more clearly, the time-averaged network-relative dispersions as a function of drop size are plotted in Fig. 13.
The observed values increase with increasing network
size (area) and with increasing drop size at least up to 2–
3-mm diameters. Beyond that, the average relative dispersion appears to decline. This is somewhat surprising
given the earlier discussion above (i.e., increased
patchiness and lower mean probabilities at larger sizes).
To explore why this might be happening, we take the
simulation results and compute the network-averaged
RD over time. This yields about 2000 samples as compared to the observed 50–100 samples at the larger drop
sizes. If controlled simply by sample statistics, this would
suggest a decrease in RD by over a factor of 4. However,
the opposite occurs as illustrated by the dotted–dashed
curve in Fig. 13.
Despite the much larger samples corresponding to the
simulation, the relative dispersion actually increases
monotonically for drop sizes greater than 1-mm diameter as we would expect and in contrast to the direct
but apparently misleading observations. We believe,
then, that this difference reflects an inadequate sampling
of the physics by the set of observations. This phenomenon has been observed before, such as when investigators attempt to use a limited set of disdrometer
observations to compute the so-called Z–R relations
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JOURNAL OF THE ATMOSPHERIC SCIENCES
FIG. 13. The 1-min-average RD over the entire observation period as functions of the drop size and network size compared to that
for the expanded dataset. The rolloff at larger drop sizes is attributed to incomplete sampling by the observations as discussed in the
text.
prevalent throughout radar meteorology (Jameson and
Kostinski 2002) and to deduce correlations among parameters (e.g., Jameson 2015, Fig. 12 therein). There, it
was found that false correlations and data misfits occurred because of an extreme undersampling of the
number of drops. The effects of this undersampling of
the physics could, in part, be alleviated using a dataset
expanded along the approach outlined above (e.g.,
Jameson 2015). The observations, after all, are just
a limited set of realizations from the rich statistical
processes described more completely by the correlation
functions and distributions of mean counts for the different drop sizes. The simulation, then, provides access
to many more of the possibilities than just those sampled
in the observations. Apparently, this can be important at
times.
4. Summary
As discussed above, a drop size distribution can be
expressed as DSD 5 Nt 3 P(D) 5 Nt 3 PSD. The variability of PSD is then defined by the distribution of
probabilities P(PD j D) at each drop size and is investigated over a network of optical disdrometers. It was
found that the variability as measured by the relative
dispersions RD of P(PD j D) as functions of drop size
increases with increasing size of the network and with
increasing size of the drops. Furthermore, the variability
VOLUME 72
over a network is often significantly greater than that
implied by the intrinsic PSD—that is, the PSD deduced
from one or more instruments after excluding all zero
drop counts.
Both of these observations can be explained in terms
of the spatial spectrum of P(D). That is, each component
of such a spectrum contributes to the variance and to the
RD. As a network size (area) increases, more spectral
components contribute, thus increasing RD since the
average P(D) remains fixed. Another way to think of it is
that the network acts to integrate the Fourier transform
of the spatial correlation function from the smallest toward larger wavelengths as the dimension of the network increases. Likewise, the spatial distribution of
raindrops becomes spikier or patchier as the drop size
increases. From classic Fourier transform theory, this, in
turn, implies a greater number of spatial spectral components in order to characterize this spatial distribution
so RD increases with increasing drop size. These observations, then, suggest that if one has access to a characterization of the spatial spectrum of the rain (i.e., the
Fourier transform of the spatial correlation function), it
should be possible to apply an intrinsic drop size distribution properly over an area. In the absence of that information, however, the intrinsic PSD and RDi will
underestimate the true areal variability. Of course, it
must also be remembered that time averaging tends to
reduce the variability due to sampling fluctuations, but it
does not necessarily eliminate or smooth out the true
physical variability (e.g., Jameson et al. 2015).
Finally, as observed in previous studies, care must be
taken that the physics is being properly sampled. Even
though 1-min measurements over 19 detectors for
440 min sounds like a lot of data, it was found that even
that amount of information did not provide a full characterization of the likely true variability of the PSD
without additional expansion of the set of observations
because the observations themselves represent only one
realization of multiple stochastic processes. It must also
be remembered that in this work we have ignored the
variability of the total number of drops, which also
contributes significantly and probably dominantly to the
variability of the rainfall rate and other integrated parameters as previously recognized by Jameson and
Kostinski (2001).
Acknowledgments. This work was supported by the
National Science Foundation (NSF) under Grant
AGS130087 as well as by the United States Social Security Administration. Support for ML came from the
National Science Foundation under Grant AGS1230240. Support for AK came from NSF Grant AGS111916. The authors are also especially grateful to the
APRIL 2015
JAMESON ET AL.
students of Prof. Larsen, namely Joerael Harris, Robert
Lemasters, Katelyn O’Dell, Joshua Teves, and Michael
Chute who diligently worked to make this network
a functioning reality.
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